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Approximate Counting for Spin Systems on Planar Graphs

Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting $q$-colourings on planar graphs is NP-hard for every constant $q \geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.

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theoretical-computer-scienceAug 6, 2026Significance 18/100Registry: unreviewed

Approximate Counting for Spin Systems on Planar Graphs

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Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting $q$-colourings on planar graphs is NP-hard for every constant $q \geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.

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Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting $q$-colourings on planar graphs is NP-hard for every constant $q \geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.

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